Math
Spring Operational 2015
Algebra 2
PBA Item #16
Broadcast Frequency
3052-M44168
Prompt
Rubric
Task is worth a total of 6 points
3052-M44168 Rubric Part A
Score
3
Description
Student response includes the following 3 elements.
•
•
Modeling component = 2 points
o
Valid model for the new area
o
Valid model for the difference in areas
Computation component = 1 point
o
Correct original area
Sample Student Response:
2
πr , so the radio tower could previously
The area of a circle is A =
reach an area of A1 =
π (20 )
2
400π square miles.
=
The area of land that the radio tower can now reach depends on how
much the radius increased. Let x be the number of miles that the
radius increased. The area of land that the tower’s signals can now
reach is A2 =
π(20 + x )
2
(
=
π x
2
)
+ 40x + 400 square miles.
To find the difference in the areas, subtract the two areas:
(
A2 − A1 =
π x
(
=
π x
2
)
2
)
+ 40x + 400 − 400π
+ 40x square miles
(or equivalent model)
Note: All three components may be presented together in a single
equation or expression. The models do not need to be identified.
2
Student response includes 2 of the 3 elements.
1
Student response includes 1 of the 3 elements.
0
Student response is incorrect or irrelevant.
3052-M44168 Rubric Part B
Score
3
Description
Student response includes the following 3 elements.
•
•
Modeling component = 2 points
o
Correct work
o
Correct explanation
Computation component = 1 point
o
Correct answer
Sample Student Response:
The area of land that the radio station is able to reach after the
radius has been increased is represented by A =
π (20 + x )
2
square
miles.
Since there are about 74 people per square mile, the radio station is
able to reach a population, P, of about
=
P
74 • A
2
P = 74π (20 + x ) people
It is given that this is equal to 160,000 people, so we can substitute
to solve for x.
160, 000 = 74π (20 + x )
2
74π (20 + x )
160, 000
=
74π
74π
=
688.2375917
(20
(20
=
688.2375917
=
26.23428276
+ x)
2
2
+ x)
2
20 + x
6.23428276 = x
2
This means that the radius that the radio tower’s broadcast signals
can reach increased by approximately 6.23 miles.
Note: The acceptable range for the third element [correct answer] is
6.22–6.25 miles.
Student response includes 2 of the 3 elements.
1
Student response includes 1 of the 3 elements.
0
Student response is incorrect or irrelevant.
Anchor Set
A1 – A14
A1
Part A: Score Point 3
Part B: Score Point 3
Annotations
Anchor Paper 1
Part A: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
•
•
The student correctly found the original area (400𝜋).
The student created an appropriate model for the new area ((20 + 𝑥)2 𝜋).
The student created an appropriate model for the difference in areas
(a = ( (20 + x)2 π) − (400π)).
The expression for the model contains a single variable (𝑥), which is the increase in radius
and does not need to be defined. The variable representing area (𝑎) does not need to be
defined.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student found the correct answer (6.23).
•
The student showed correct work toward finding the answer. It is not necessary to
subtract 20 in the work.
•
The student correctly explained the answer (the radius increased by 6.23 miles).
A2
Part A: Score Point 3
Part B: Score Point 3
Annotations
Anchor Paper 2
Part A: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
•
•
The student correctly found the original area (400𝜋).
The student created an appropriate model for the new area ((20 + 𝑥)2 𝜋).
The student created an appropriate model for the difference in areas (f(x) = (20 +
x)2 π − 400π).
The expression for the model contains a single variable (𝑥), which is the increase in radius
and does not need to be defined. The function representing area (𝑓(𝑥)) does not need to be
defined.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student found the correct answer (6.23).
•
The student showed correct work toward finding the answer
•
The student correctly explained the answer (The radio station increased its broadcast
radius by 6.23 miles).
The first line of calculations is actually broken into two lines; when it is seen as a single line,
squaremile
� = 2162 square miles).
it is correct (total number of square miles: (160,000people) �
74people
A3
Part A: Score Point 2
Part B: Score Point 3
Annotations
Anchor Paper 3
Part A: Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
The student correctly found the original area (400𝜋).
•
The student created an appropriate model for the new area (𝑥 2 𝜋 + 40𝑥𝜋 + 400𝜋).
No model is presented to show the difference in areas.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student found the correct answer (6.24) [within the acceptable range of 6.22–
6.25].
•
The student showed correct work toward finding the answer.
•
The student correctly explained the answer (so the new radius is 26.24 miles. After
subtracting the original 20 miles, one gets 6.24 miles. This is the miles that the radio
station increased their broadcast radius) [any of these three sentences would be a
sufficient explanation].
A4
Part A: Score Point 3
Part B: Score Point 2
Annotations
Anchor Paper 4
Part A: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student correctly found the original area, created an appropriate model for the
new area, and created an appropriate model for the difference in areas ((40 + 𝑥)𝜋 ∙ 𝑥).
This particular model receives credit for all three elements; the new area and initial
area are rewritten and contained within the equation.
Part B: Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
The student found the correct answer (6.23).
•
The student showed correct work toward finding the answer.
No credit is given for the explanation (𝑥 = 6.23 mi); without more explanation, the answer
must be labeled as the increase in miles.
A5
Part A: Score Point 2
Part B: Score Point 2
Annotations
Anchor Paper 5
Part A: Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
•
The student correctly found the original area ((202 × 𝜋)).
The student created an appropriate model for the new area ((20 + 𝑥)2 × 𝜋)).
No credit is given for the difference in areas: while the left side of the equation is correct, it is
set equal to the difference in radius, which is incorrect.
Part B: Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
The student found the correct answer (6.23).
•
The student correctly explained the answer (6.23 miles was how much it widened its
radius).
No credit is given for work: while the work is described verbally, the description is incorrect
(dividing 160,000 by 74. Then you divide by pie and subtract 20 from youre answer). The
description should have included taking the square root before subtracting 20. It is not
necessary to describe every single step, but in this case, the response moves directly from
dividing by pi to subtracting 20.
A6
Part A: Score Point 1
Part B: Score Point 3
Annotations
Anchor Paper 6
Part A: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student correctly found the original area (𝜋400).
No credit is given for the new area (𝜋𝑟 2 ) or for the difference in areas. Because the prompt
specifies that the variable (𝑟) must represent the increase in the radius, the models are
incorrect.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student found the correct answer (6.23).
•
The student showed correct work to find the answer.
•
The student correctly explained the answer (6.23 radius increased).
A7
Part A: Score Point 3
Part B: Score Point 0
Annotations
Anchor Paper 7
Part A: Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
The student correctly found the original area (400π).
•
The student created an appropriate model for the new area((20 + 𝑥)2 𝜋).
•
The student created an appropriate model for the difference in areas((20 + 𝑥)2 𝜋 −
400𝜋).
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the answer: an incorrect answer was found (905).
No credit is given for the work or explanation: an incorrect process was shown and explained
(I divided 74 by 160,000 so I could find out how many square miles the radio now
broadcasted. Next, I subtracted how much the radio broadcasted before . . . and got my
answer of approximately 905 square miles), describing the difference in areas as the
difference in radius.
A8
Part A: Score Point 1
Part B: Score Point 2
Annotations
Anchor Paper 8
Part A: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student correctly found the original area (𝜋 × 202 ).
No credit is given for the new area (𝜋 × 𝑟 2 ) or for the difference in areas. Because the prompt
specifies that the variable (𝑟) must represent the increase in the radius, the models are
incorrect.
Part B: Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
The student showed correct work to find the answer, found the number of people the
station originally reached, then found the ratio of the number of people the station
could finally reach to the number of people it originally reached (1.72), and multiplied
that value times the original area to find the new area (2161.4036); the new area was
divided by pi, the square root was taken, and 20 was subtracted.
•
The student correctly explained the answer (6.38 miles increased).
No credit is given for the answer (6.38): the value is outside the acceptable range of 6.22–
6.25.
A9
Part A: Score Point 2
Part B: Score Point 0
Annotations
Anchor Paper 9
Part A: Score Point 2
This response receives full credit. The student includes each of the three required elements:
•
The student correctly found the original area (𝜋202 = 1256.637) [either would be
credited].
•
The student created an appropriate model for the new area (𝜋(20 + 𝑋)2 ).
No model was presented to represent the difference in areas.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the answer: an incorrect answer was found (905.525).
No credit is given for work: the work shows an incorrect process to find the answer
[subtracting the old area from the new]. These could have been the first steps in a correct
process, if the answer had then been set equal to the model for the new area from Part A,
but because the work stopped at this point, it is considered incorrect.
No explanation was presented.
A10
Part A: Score Point 1
Part B: Score Point 1
Annotations
Anchor Paper 10
Part A: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student correctly found the original area (400𝜋).
No credit is given for the new area (𝜋𝑥 2 ) or for the difference in areas. Because the prompt
specifies that the variable (𝑥) must represent the increase in the radius, the models are
incorrect.
Part B: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student showed correct work toward finding the answer. It is not necessary to
subtract 20 in the work.
No credit is given for the answer: an incorrect answer was found (26.23).
No explanation was presented. The label on the answer (26.23 miles) is not credited; without
more explanation, this particular answer must be labeled as the length of the new radius for
credit.
A11
Part A: Score Point 1
Part B: Score Point 0
Annotations
Anchor Paper 11
Part A: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student correctly found the original area (A = 3.14(202 ); A = 1256.6) [either of these
would be credited].
No credit is given for the new area: no model was presented.
No credit is given for the difference in areas: the difference was described (area of increased
radius minus the original radius equals the square miles of the new area) but no model was
presented.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the answer (approximately 6): an answer of 6 must be supported by
some work.
No work or explanation was presented.
A12
Part A: Score Point 0
Part B: Score Point 1
Annotations
Anchor Paper 12
Part A: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the original area (𝜋𝑟 2 ): if a variable is used for the original radius, it
must be defined as 20.
No credit is given for the new area (𝜋(𝑟 + 𝑥)2 ): while the variable for the increase in area was
defined (Let x be the increase in radius), the variable r was not defined and the value of r
was not given.
No credit is given for the difference in areas: the variable r was not defined and the value of r
was not given. The variable x is also used in two different places, making the equation
incorrect.
Part B: Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
The student correctly explained the answer (If you raise the radius by 6.3 miles, you
have an area of 2173 miles. With 74 people per square mile, you get approximately
160,000 people. The radio station increased its broadcast radius by 6.3 miles).
No credit is given for the answer (6.3): the value is outside the acceptable range of 6.22–
6.25.
No credit is given for work: the work that is described is guess-and-check, giving one guess
for the answer (6.3) and checking that that value will produce the correct result. With the
guess-and-check method, at least two values must be checked.
A13
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 13
Part A: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response (𝑟 = 𝑟 + 𝑚) does not match any of the models.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the answer: an incorrect answer was found (26).
No credit is given for work: the work shown is incorrect (16000 ÷ 𝜋 ≈ 688 )[it approximately
equals 5093]
No explanation was presented.
A14
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 14
Part A: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response (𝐴 = 2𝜋𝑟 − 2𝜋 ∙ 20) does not match any of the models. It gives the difference
between the circumferences of a circle of radius r and a circle of radius 20.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
No credit is given for the answer: an incorrect answer was found (2162.16).
No credit is given for work: insufficient work toward finding the answer was shown [the new
area was calculated].
No explanation was presented.
Practice Set
P101 - P105
P101
P102
P103
P104
P105
Practice Set
Paper
Score
P101
3,3
P102
1,0
P103
2,0
P104
3,2
P105
1,2